Research On Golden Ratio Algorithm And Subgradient Extragradient Algorithm For Equilibrium Problem

Equilibrium problem is an attractive and hot research subject of nonlinear analysis and optimization theory,and it has extremely broad application prospects extremely extensive application prospect in different fields,such as applied mathematics,management science,ecological economics,physics and operation research.Equilibrium theory is closely related to fixed point theory,game theory,variational inequality,optimization and differential equation,which provides an important tool for the development of these theories.Since the equilibrium problem is of great significance in the optimization theory,it has received extensive attention and in-depth research from scholars at home and abroad in the theory and algorithm of the equilibrium problem.One of the most popular research directions of equilibrium problem is how to construct an effective iterative algorithm,find the approximate solution of equilibrium problem,and analyze the convergence of the proposed algorithm.In the process of constructing an effective iterative algorithm for solving equilibrium problems,the selection of step size is essential.By constructing an effective step size criterion,this paper presents an effective numerical algorithm for solving the balance problem,and analyzes the convergence of the proposed algorithm.This paper avoids the technical shortcomings of using Lipschitz-type constants or Armijo criteria to set the step size in the classic algorithm.Combined with the non-monotonic step size selection strategy,this paper introduces two new types of numerical algorithms for solving the balance problem,and analyzes the convergence of the algorithm.The algorithms given can be regarded as a further study on the basis of two classic algorithms,the golden ratio algorithm and the subgradient extragradient gradient algorithm.The specific work is as follows:Firstly,a modified golden ratio algorithm for the equilibrium problems is proposed.Inspired by Malitsky's golden ratio algorithm,a new and concise algorithm is designed.The proposed algorithm only needs to calculate a strong convex programming problem per iteration,and the selection of the step size does not depend on the Lipschitz-type constants.Under the condition that the bifunction is pseudomonotone,it is proved that the modified algorithm weakly converges to the solution set,and under suitable conditions,the R-linear convergence rate is established.In particular,the introduced algorithm is applied to the corresponding variational inequality problems.Numerical experiments show the effectiveness of the algorithm.Secondly,an inertial subgradient extragradient algorithm for the equilibrium problems is proposed.Inertial method is a method to accelerate the convergence of algorithms in recent years.Using a new step size rule,a subgradient extragradient algorithm with inertial effect is proposed to find the common solution of the equilibrium problem and the fixed point problem.Under the reasonable assumption of bifunction,the weak convergence of the proposed algorithm is established.In addition,the corresponding variational inequality problem is also considered.Finally,numerical experiments are used to prove the feasibility and superiority of the algorithm.